The covering and filling factors strongly depend on the poorly known viscosity, parametrized with d. Furthermore, both f S and f V depend on high powers of another poorly known parameter, the stripping radius R 0.
Hence, properties of the subcluster wakes can be rather different in apparently similar clusters. In addition, results of numerical simulations of turbulent wakes should be treated with caution as otherwise reasonable approximations, numerical resolution, and numerical viscosities can strongly affect the results. The size of galactic wakes required to cover the projected cluster area, given by equation 16 , does not seem to be unrealistic.
For example, Sakelliou et al. These authors argue that the wake is produced by the ram pressure stripping of the interstellar gas. The projected area of the wake is about 10 4 kpc 2 , as compared to 10 3 kpc 2 for the wake parameters derived above. This wake has been detected only because it is exceptionally strong, and it is not implausible that weaker but more numerous galactic wakes can cover the area of the central parts of galaxy clusters. We conclude that subcluster wakes are likely to be turbulent, but galactic wakes can be laminar if the viscosity of the intracluster gas is as large as Spitzer's value.
Given the uncertainty of the physical nature and hence, estimates of the viscosity of the magnetized intracluster plasma, we suggest that turbulent galactic wakes remain a viable possibility. Both types of wake have low volume-filling factor but can have an area-covering factor of the order of unity. In this section, we discuss the amplification of an initially weak seed magnetic field by the fluctuation dynamo operating in the intracluster gas.
The seed field itself can be produced by a wide range of mechanisms Appendix A ; see also Ruzmaikin et al. We first discuss the fluctuation dynamo in general terms. These general ideas are then applied to the various contexts of intracluster turbulence discussed above. First, we consider the merger epoch when the turbulence can be assumed to be in a statistically steady state, then the later epochs after the driving by the merger had ceased and the turbulence decays, and finally to magnetic field generation in turbulent wakes.
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The exponentially fast amplification of an initially weak magnetic field by a random flow called the fluctuation dynamo is a result of a random stretching of magnetic field by the local velocity shear see reviews in Zeldovich et al. Such amplification comes at the cost of a decrease in the scale of field structures in the directions perpendicular to the stretching i. This enhances Ohmic dissipation and the latter ensures that the correlation function of magnetic field can grow exponentially as an eigenfunction if the Lorentz force is negligible the kinematic dynamo.
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Such simulations are also able to follow the fluctuation dynamo into the non-linear regime where the Lorentz force becomes strong enough to affect the flow as to saturate the growth of magnetic field. In a turbulent flow, where a broad spectrum of motions is present, flow at each scale of l would produce magnetic structures at all scales down to the corresponding Ohmic scale.
However, in the non-linear regime, when the fastest growing mode saturates, larger-scale modes could still grow. Since most of the kinetic energy is contained at the scale of l 0 , the dominant magnetic scale could still be determined by dynamo action due to eddies of the scale of l 0 and, especially, by the subtle details of the dynamo saturation. We now discuss how the dynamo action could saturate. Such results, however plausible they are, require further substantiation, for example, by numerical simulations.
Dynamo simulations of Haugen et al. We note, however, that it is not quite clear how significant is the difference or agreement here since all the estimates have factors of the order of unity omitted, which can be important at the modest values of R m available. Further, these simulations also show that the value of k B does not scale with R m when R m is increased from about to , confirming equation However, the magnetic spectrum is rather broad and it is difficult to identify accurately the dominant magnetic scale in those simulations.
Nevertheless, it is clear that the non-linear magnetic field distribution is less intermittent i. Below we present evidence for this non-linear behaviour in our own simulations of the fluctuation dynamo. The values of magnetic Reynolds number accessible now in such simulations are too modest to make any confident conclusions, but we believe that our approach to the saturation of the fluctuation dynamo is consistent with the evidence available. Nevertheless, in both the simulations presented below and in real clusters the flow is not strongly turbulent, that is, it has no extended Kolmogorov inertial range because the Reynolds number is not very large; so equation 19 can remain a better approximation.
There is some evidence for this from the simulations, in that equation 19 agrees better with the wave number at which the magnetic spectrum peaks. We will therefore use equation 19 in our estimates. Here we present semi-quantitative estimates to characterize the fluctuation dynamo in the intracluster gas, before discussing, in Section 4 , direct numerical simulations of the fluctuation dynamo. Here , so each frame contains a few turbulent cells. The magnitude of the field component perpendicular to the plane of the figure is shown colour coded in shades of grey with black corresponding to field pointing into the figure plane, and lighter shades, to field pointing out of the plane.
The field in the plane of the figure is shown with vectors whose length is proportional to the field strength. However, this quantity has little physical significance because it would not result from any local magnetic measurement. In this sense, the local value 24 is more meaningful; it is presented in Table 1 together with other quantities that characterize turbulence and magnetic fields at various stages of the cluster evolution.
We have simulated the generation and subsequent decay of dynamo-active turbulence using the numerical model of the fluctuation dynamo by Haugen et al. The Navier-Stokes, continuity and induction equations are then solved in a Cartesian box of a size D on a cubic grid with 3 mesh points. The direction of the wave vector of the force and its phase change randomly every time step in the simulations, so the force is effectively d-correlated in time. We represent numerical results using the following units.
Tilde is used to denote dimensionless quantities.
Then the unit length is and the dimensional size of the computational domain is. Here, we report results obtained with two values of the central driving wave number k f. Some results were obtained with driving covering the range of dimensionless wave numbers , centred at Model 1. In these latter runs, the computational box contains just a few turbulent cells. The intensity of the driving was adjusted to obtain the rms Mach number of the turbulence of about 0.
We note, however, that the values of magnetic Reynolds number explored here still are smaller than those expected in reality. In order to simulate dynamo action in forced and then decaying turbulence, the flow had been driven until it reached a statistically steady state, with a weak magnetic field introduced at the start of the simulation.
The initial, weak magnetic field is random, with energy density of about 0. The initial exponential growth of the rms magnetic field that obtains not shown in Fig. More precisely, the rms values of the turbulent velocity and magnetic field measured in velocity units in the steady state of Fig. The critical value of the magnetic Reynolds number remains about 35 in both models.
Hence, v rms numerically coincides with Re in the statistically steady state. Time is measured in the units of the initial turnover time of the energy-containing eddies, t oi. The magnetic and kinetic energy spectra are shown in Fig. In the statistically steady state the upper curves , kinetic energy in Model 1 peaks at , the driving wave number. However, magnetic energy has broad maximum at a significantly smaller scale, apparently because of its intermittent structure. This difference is better visible in the right-hand panel that refers to Model 2 where we have higher resolution.
Similar simulations, but with a significantly higher resolution 3 Haugen et al. Spectra obtained at later times are at lower levels because of the decay of turbulent energy. Evolution of the integral scales of the velocity L v , solid line and magnetic fields L B , dashed line , as defined in equation 25 , with Model 1.
Magnetic energy at small scales has, at early times, excess over kinetic energy because magnetic field is very intermittent, which is, especially, clearly visible in the right-hand panel of Fig. At later stages, magnetic field distribution becomes more homogeneous and this feature disappears.
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Simultaneously, the scale of magnetic field increases and becomes comparable to that of the flow, which is not the case at early stages. The magnetic field produced by the fluctuation dynamo consists of an intermittent part, represented by randomly distributed, intense magnetic ribbons, sheets and filaments which can even be folded , immersed in a sea of volume-filling random magnetic field.
The intermittency gradually reduces as the turbulence decays together with magnetic field because structures of smaller scale decay faster, and the volume-filling factor of magnetic field increases with time—this tendency can easily be seen in the right-hand panel of Fig. It is useful to compare results of the simulations with the analytical estimate of equation 28 , in dimensionless units used in Fig.
This gives for in Model 2 and for in Model 1. This estimate of for Model 2, which has a higher spatial resolution, is in good agreement with the numerical simulations, but that obtained for Model 1 is a factor of about 2 lower than expected.
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Nevertheless, our simulations confirm that equation 28 and Table 1 provide reasonably good estimates of the expected amount of Faraday rotation by magnetic field generated by the fluctuation dynamo. Altogether the estimate 28 and the amount of Faraday rotation in our simulations agree very well with observations of Faraday rotation in the intracluster gas. As turbulence decays and magnetic field becomes less structured, the correlation scale of RM fluctuations increases.
The former curve refers to the statistically steady state, whereas the latter two illustrate how RM distribution becomes less intermittent as turbulence decays. Here R is measured in the units of. Estimates of typical Faraday RMs obtained from equations 28 and 30 are given in the last column of Table 1. Schekochihin et al. However, observations indicate that magnetic coherence scale is at least a few kpc and more plausibly exceeds 10 kpc.
This would be difficult to produce in the model of Schekochihin et al. Furthermore, Schekochihin et al. We have also used numerical simulations to examine the effects of varying the magnetic Prandtl number on magnetic field structure and Faraday rotation. The probability distribution of the Faraday RM along 2 lines of sight through the computational box is shown in Fig.
This implies that magnetic field does not become more strongly folded as P m increases. As in Fig. We emphasize again that both the form of the RM probability distribution and the correlation function do not change much as P m increases from unity to Nevertheless, it would be important to clarify this issue further using simulations with higher resolution.
Shear in the gas motions at a scale of a few hundred kpc, produced during major merger events, can make the random magnetic field locally anisotropic.
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